If \(f:X\rightarrow Y\)  is a \(K\) -morphism, we get a pullback functor \(f^*:_n(Y)\rightarrow _n(X)\) , so that \(_n\)  is a category fibered in groupoids.


We will show that \(_n\)  is actually representable by a perfectoid space, and therefore a stack for the analytic topology.


Note that if some \(\alpha _i\)  exists, it is unique. Indeed, for each \(i\)  the global sections \(s_j^{(i)}\)  generate \(_i\) , so that an isomorphism \(_{i+1}^{\otimes p}_i\)  shows that the global sections \(\left(_{j}^{(i+1)}\right)^{\otimes p}\)  generate \(_{i+1}^{\otimes p}\) . In particular, the isomorphism is completely determined by the images of these global sections.


For each \(i\) , the data \(\left(_i,s^{(i)}_j\right)\)  corresponds to a map to a projective space (as a rigid analytic variety), so that objects of the category \(_n(X)\)  correspond to \(p\) th power root systems of maps to projective space.

The main result of this section is that the category \(_n(X)\)  parametrizes \(K\) -morphisms \(X^{n,}\) . In particular, viewing \(_n\)  as a functor to sets we construct a natural isomorphism \((\cdot ,^{n,})\cong _n\)  of functors from perfectoid spaces over \(K\)  to sets. First we introduce a bit of notation.

Denote by \(m_i:(1/p^{i+1})^{\otimes p}{\sim }(1/p^i)\)  the isomorphism of line bundles on \(^{n,}\)  coming from multiplying factors together.

We now state the main theorem of this section (compare to [1]} Theorem II.7.1).

The functor \(_n\)  is represented by projectivoid space.
